Scaling Dynamics of Quantum Turbulence from a Box Trap

The large-scale expansion of quantum gases is a widely used fundamental technique, providing key insights into momentum distributions in ultracold gas research. In this study, we investigated the evolution of the system size of 3D turbulent states initially trapped in box potentials using numerical methods, primarily leveraging the Fast Fourier Transform (FFT) method for efficient derivative calculations of a reformulated Gross-Pitaevskii treatment. Comparisons between numerical solvers reveal that the adaptive-timestep ODE solver in Julia, which uses a 6^th Order Runge-Kutta method algorithm, significantly outperforms the fixed-timestep Fourth-Order Runge-Kutta method (RK4) in computational efficiency for larger systems. The advantage of our approach is that we could follow the system in a fine spatial grid through long expansion time.

When the initial excitation of turbulence (characterised by the forcing amplitude U_F) is small, our results demonstrate that free expansion leads to the well-known dynamics in free space. Also, when U_F increases, the energies of the states during free expansion approach universal time evolution in the high energy regime. For turbulent states with higher energy, the density evolves to be more isotropic after long-time expansion, with a residual weak dependence on the initial isotropicity. This is consistent with an increasingly isotropic momentum distribution at higher energies, associated with a well-developed turbulent state. In addition, our results also show a rapid growth in number of vortices as U_F increases beyond a characteristic trap energy. An advantage of long-time free expansion is that the numerical resolution of momentum space becomes very high. We compared a semi-analytical spectral analysis method with the traditional numerical binning method in the study of momentum densities in k-space, finding that the method of spectral analysis gives a more detailed and reliable description of momentum densities, with clear advantages for analysis at lower k values and smaller U_F. Additionally, in an antitrapping potential, scaling parameters characterising the system size exhibit exponential growth creating spatial state magnification for any initial states, while freezing all other dynamics. The numerical results for the scaling parameters and energy evolution of the states in antitrapping system show strong agreement with analytical solutions. Our results also provide a deeper insight into energy decay and vortex dynamics in this system.